Educational interventions are effective tools for enhancing student learning. While Large Language Models (LLMs) allow for generating adaptive feedback at scale, current studies lack clear methodologies for providing Just-in-Time (JiT) feedback in authentic instructional settings. In this paper, we present a framework that provides adaptive feedback by grounding LLMs with domain-specific expert knowledge. Our approach collects written reasoning logic (strategy essays) from students, analyzes potential error types based on the content of that reasoning, and delivers non-intrusive feedback designed to clarify missing or incorrect concepts. We deploy this framework in a large-scale university course (N > 1000), where it improved student performance by over 80% compared to previous semesters. Lastly, we validate the framework's pedagogical utility by analyzing the learning trajectories; we demonstrate how iterative conversations with LLM facilitate shifting one's misconception to correct understanding.